Problem 12 - University Physics with Modern Physics

Question

Suppose you are explaining the theory of relativity to a friend, and you have told him that nothing can go faster than \(300,000 \mathrm{~km} / \mathrm{s}\). He says that is obviously false: Suppose a spaceship traveling past you at \(200,000 \mathrm{~km} / \mathrm{s}\), which is perfectly possible according to what you are saying, fires a torpedo straight ahead whose speed is \(200,000 \mathrm{~km} / \mathrm{s}\) relative to the spaceship, which is also perfectly possible; then, he says, the torpedo's speed is \(400,000 \mathrm{~km} / \mathrm{s}\). How would you answer him?

Step-by-Step Solution

Answer

In the given scenario of a spaceship and a torpedo, the theory of relativity takes into account the combination of their velocities and ensures that the total velocity does not exceed the speed of light (\(300,000 \mathrm{~km} / \mathrm{s}\)). According to special relativity's velocity addition formula, the relative velocity of the torpedo with respect to the observer is approximately \(327,273 \mathrm{~km} / \mathrm{s}\), which is less than the speed of light. Therefore, there is no contradiction, and the student's misconception is addressed.

Step 1: Introduce the Principle of Relativity

The theory of relativity states that the laws of physics are the same for all inertial observers, regardless of their relative velocities. In this case, we have a spaceship and a torpedo that are moving relative to an observer.

Step 2: Explain Velocity Addition in Classical Mechanics

In classical mechanics, velocities simply add together. According to this framework, if the spaceship is traveling at \(200,000 \mathrm{~km} / \mathrm{s}\) and the torpedo is traveling at \(200,000 \mathrm{~km} / \mathrm{s}\) relative to the spaceship, the student is correct that the total speed would be \(400,000 \mathrm{~km} / \mathrm{s}\), which appears to contradict the theory of relativity.

Step 3: Explain Velocity Addition in Special Relativity

However, special relativity provides a different equation for combining velocities, accounting for the fact that nothing can travel faster than the speed of light (\(300,000 \mathrm{~km} / \mathrm{s}\)). The relativistic velocity addition formula is given by: \(v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\), where \(v\) is the relative velocity between the two objects, \(v_1\) and \(v_2\) are their individual velocities, and \(c\) is the speed of light.

Step 4: Calculate the Relative Velocity of the Torpedo

Using the given example, let's calculate the relative velocity of the torpedo (in km/s) with respect to the observer: \(v_1 = 200,000 \mathrm{~km} / \mathrm{s}\) (spaceship velocity) \(v_2 = 200,000 \mathrm{~km} / \mathrm{s}\) (torpedo velocity relative to the spaceship) \(c = 300,000 \mathrm{~km} / \mathrm{s}\) (speed of light) Plug these values into the relativistic velocity addition formula: \(v = \frac{200,000 + 200,000}{1 + \frac{200,000 \cdot 200,000}{300,000^2}}\) \(v = \frac{400,000}{1 + \frac{40,000,000,000}{90,000,000,000}}\) \(v = \frac{400,000}{1 + \frac{2}{9}}\) \(v = \frac{400,000}{\frac{11}{9}}\) \(v = \frac{400,000 \cdot 9}{11}\) \(v \approx 327,273 \mathrm{~km} / \mathrm{s}\)

Step 5: Conclusion

The relative velocity of the torpedo with respect to the observer is approximately \(327,273 \mathrm{~km} / \mathrm{s}\), which is less than the speed of light, \(300,000 \mathrm{~km} / \mathrm{s}\). Thus, the theory of relativity holds true and the student's misconception is cleared. There is no contradiction, and no object can travel faster than the speed of light according to the theory of relativity.

Key Concepts

Velocity Addition

In everyday situations that we encounter, adding velocities seems straightforward. If a car is moving at 60 km/h and it passes another object moving at 20 km/h in the same direction, we often consider the object to be moving at 80 km/h relative to the ground. This is known as the classical concept of velocity addition. However, in the realm of high speeds close to that of light, this additive approach does not give accurate results. For objects moving at velocities significant compared to the speed of light, the laws of physics, particularly those stated in Einstein's theory of special relativity, provide a more accurate method to add velocities. This is where the concept of relativistic velocity addition comes into play.

Speed of Light

The speed of light, denoted by the symbol \( c \), is a fundamental constant in physics, with a value of approximately 300,000 kilometers per second. This constant is crucial when discussing relativistic physics, as it serves as the ultimate speed limit for the transmission of information and energy. No object with mass can reach or exceed this speed, as it would require infinite energy. According to Einstein's theory of relativity, the speed of light is always constant regardless of the observer’s frame of reference or the motion of the light source. This principle is one of the foundational aspects of special relativity and dramatically changes how we understand motion and velocities at high speeds.

Inertial Observers

When discussing the theory of relativity, it is essential to understand the concept of inertial observers. An inertial observer is someone who is either at rest or moving at a constant velocity, meaning that they are not accelerating. In the framework of special relativity, all the laws of physics are the same for every inertial observer. This implies that no matter how fast an object is moving (as long as it is constant), the speed of light remains the same, illustrating a key aspect of Einstein’s theory. Understanding how different observers perceive the same events is crucial in analyzing scenarios involving high velocities and the addition of velocities under relativistic conditions.

Relativistic Velocity Formula

To address the complexities of high-speed motion, special relativity introduces a formula for combining velocities that differs from simple addition used in classical mechanics. This formula is termed the relativistic velocity addition formula and is written as: \[v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}.\] In this formula:

• \( v \) is the resulting velocity as observed from a particular frame.
• \( v_1 \) and \( v_2 \) are the individual velocities of the two objects being considered.
• \( c \) is the speed of light.

This formula ensures that the resulting velocity \( v \) will never exceed the speed of light. Even if \( v_1 \) and \( v_2 \) are each sizeable, the term \( \frac{v_1 v_2}{c^2} \) in the denominator modifies the sum to reflect relativistic effects, illustrating how velocities behave differently as they get close to the speed of light.